Corollary (??).

39.A. Symmetric compositions and Tits constructions. In this section,

we assume that char F = 3. The aim is to show that Tits constructions with admis-

sible pairs (1, ν) are also Springer constructions. We start with a ¬rst Tits construc-

tion (A, »); let L = F [X]/(X 3 ’ ») be the cubic Kummer extension associated with

» ∈ F — , set, as usual, A0 = { x ∈ A | TA (x) = 0 } and let “(A, ») = (A0 —L, L, N, β)

be the twisted composition of type 1A2 induced by A and » (see (??)). Let

J “(A, ») = L • “(A, ») = L • A0 — L = L — A

be the Freudenthal algebra obtained from “(A, ») by the Springer construction. If

v is the class of X modulo (X 3 ’ ») in L, (1, v, v ’1 ) is a basis of L as vector space

over F and we write elements of L — A as linear combinations a + v — b + v ’1 — c,

with a, b, c ∈ A.

∼

(39.7) Proposition. The isomorphism φ : J “(A, ») ’ J(A, ») = A • A • A

’

given by

a + v — b + v ’1 — c ’ (a, b, c) for a, b, c ∈ A

is an isomorphism of Freudenthal algebras.

Proof : We use the map φ to identify L as an ´tale subalgebra of J(A, ») and get a

e

corresponding Springer decomposition

J(A, ») = L • L — A0 .

It follows from Theorem (??) and from the description of a twisted composition

“(A, ») of type 1A2 given in § ?? that φ restricts to an isomorphism of twisted

∼

compositions “(A, ») ’ L — A0 , hence the claim.

’

(39.8) Corollary. Tits constructions are Freudenthal algebras.

Proof : We assume char F = 3. By descent we are reduced to ¬rst Tits construc-

tions, hence the claim follows from Proposition (??).

(39.9) Corollary. Let G be a split simple group scheme of type F4 . Jordan alge-

bras which are ¬rst Tits constructions are classi¬ed by the image of the pointed

set H 1 (F, PGL3 —µ3 ) in H 1 (F, G) under the map PGL3 —µ3 ’ G induced by

(A, ») ’ J(A, »).

Not all exceptional Jordan algebras are ¬rst Tits construction (see Petersson-

Racine [?] or Proposition (??)). Thus the cohomology map in (??) is in general

not surjective (see also Proposition (??)).

We now show that the Springer construction associated with a twisted composi-

tion of type 2A2 is always a second Tits construction. Let (B, „ ) be a central simple

algebra with a unitary involution over a quadratic ´tale F -algebra K. Let ν ∈ K

e

be such that NK (ν) = 1; let L be as in Proposition (??), (??), and let “(B, „, ν)

be the corresponding twisted composition, as given in Proposition (??), (??).

∼

(39.10) Proposition. There exists an isomorphism L•“(B, „, ν) ’ J(B, „, 1, ν).

’

§39. THE TITS CONSTRUCTION 527

Proof : The twisted composition “(B, „, ν) over F is de¬ned by descent from the

twisted composition “(B, ν) over K (see Proposition (??)); similarly J(B, „, 1, ν) is

de¬ned by descent from J(B, ν) (see Proposition (??)). The descents are compatible

with the isomorphism given in Proposition (??), hence the claim.

39.B. Automorphisms of Tits constructions. If J is a vector space over F

with some algebraic structure and A is a substructure of J, we write Aut F (J, A)

for the group of F -automorphisms of J which maps A to A and by Aut F (J/A) the

group of automorphisms of J which restrict to the identity on A. The corresponding

group schemes are denoted Aut(J, A) and Aut(J/A).

(39.11) Proposition (Ferrar-Petersson, [?]). Let A be central simple of degree 3

and let J0 = J(A, »0 ) be a ¬rst Tits construction. The sequence of group schemes

γ ρ

1 ’ SL1 (A) ’ Aut(J0 , A+ ) ’ Aut(A+ ) ’ 1

’ ’

where γ(u)(a, a1 , a2 ) = (a, a1 u’1 , ua2 ) for u ∈ SL1 (A)(R), R ∈ Alg F , and ρ is the

restriction map, is exact.

Proof : Let R ∈ Alg F ; exactness on the left (over R) and ρR —¦ γR = 1 is obvious.

Let J0 = A+ • A1 • A2 and let · be an automorphism of J0 R which restricts to the

identity on A+ . It follows from Remark (??) that · stabilizes A1 R and A2 R , so there

R

exist linear bijections ·i : Ai R ’ Ai R such that ·(a, a1 , a2 ) = a, ·1 (a1 ), ·2 (a2 ) .

Expanding · a — (0, a1 , a2 ) in two di¬erent ways shows that

·1 (aa1 ) = a·1 (a1 ) and ·2 (a2 a) = ·2 (a2 )a.

Hence there are u, v ∈ A— such that ·1 (a1 ) = a1 v and ·2 (a2 ) = ua2 . Comparing the

R

¬rst components of · (0, 1, 1)# = ·(0, 1, 1) # yields v = u’1 . Since · preserves

the norm we have u ∈ SL1 (A)(R). To conclude, since Aut(A+ ) is smooth (see

the comments after the exact sequence (??)), it su¬ces to check by (??) that ρalg

is surjective. In fact ρ is already surjective: let φ ∈ Aut(A+ ), hence, by the

exact sequence (??), φ is either an automorphism or an anti-automorphism of A.

In the ¬rst case, φ(a, a1 , a2 ) = φ(a), φ(a1 ), φ(a2 ) extends φ to an element of

Aut(J0 , A+ ). In the second case, A is split, so some u ∈ A— has NA (u) ∈ F —2 and

φ(a, a1 , a2 ) = φ(a), φ(a2 )u’1 , uφ(a1 ) extends φ.

√ √

Now, let L = F ( 3 »). We embed L = F (v), v = 3 », in J(A, ») = A • A • A

through v ’ (0, 1, 0) and v ’1 ’ (0, 0, 1). Furthermore we set

(A— — A— )Det = { (f, g) ∈ A— — A— | NrdA (f ) = NrdA (g) }.

(39.12) Corollary. (1) We have AutF J(A, »), A+ = (A— — A— )Det /F — , where

F — operates diagonally, if A is a division algebra and

Det

AutF J(A, »), A+ = GL3 (F ) — GL3 (F ) /F — Z/2Z

if A = M3 (F ). The action of Z/2Z on a pair (f, g) is given by

(f, g) ’ (f ’1 )t , (g ’1 )t .

The action of (f, g) on J(A, ») is given by

(f, g)(a, b, c) = (f af ’1 , f bg ’1 , gcf ’1 )

and the action of Z/2Z by „ (a, b, c) = (at , ct , bt ).

528 IX. CUBIC JORDAN ALGEBRAS

(2) We have AutF J(A, »), A+ , L AutF (A)/F — — µ3 if A is a division algebra

and

AutF J(A, »), A+ , L Z/2Z

PGL3 (F ) — µ3

where the action of Z/2Z is given by „ (f, µ) = ([f t ]’1 , µ’1 ) if A = M3 (F ).

Proof : (??) If φ ∈ AutF J(A, »), A+ restricts to an automorphism φ of A, we

write φ = Int(f ) and (??) follows from Proposition (??). If φ restricts to an

anti-automorphism φ of A, we replace φ by φ —¦ „ and apply the preceding case.

(??) We assume that A = M3 (F ). By (??) we can write any element φ of

AutF J(A, »), A+ , L as [f, g] with f , g ∈ GL3 (F ). Since φ restricts to an auto-

morphism of L, we must have φ(u) = ρu±1 , ρ ∈ F — . Since „ (u) = u’1 , we may

assume that φ(u) = ρu (replace φ by φ„ ). It follows that ρ3 = 1 and ρ ∈ µ3 (F ).

Since φ (0, 1, 0) = (0, f g ’1 , 0) = (0, ρ’1 , 0) we get g = ρf with ρ ∈ µ3 (F ). The

map (f, ρ) ’ (f, ρf ) then induces the desired isomorphism.

(39.13) Remark. If F contains a primitive cubic root of unity, we may identify

Z/2Z with PGL3 (F ) S3

µ3 with A3 (as Galois-modules) and PGL3 (F ) — µ3

where S3 operates through its projection on Z/2Z. In particular we get for the split

Jordan algebra Js

AutF Js , M3 (F )+ , F — F — F = PGL3 (F ) S3 .

On the other hand we have

AutF (Js , F — F — F ) = Spin8 (F ) S3

(see Corollary (??)), so that

Det

/F — Z/2Z © Spin8 (F )

PGL3 (F ) S3 = GL3 (F ) — GL3 (F ) S3 ‚ G(F )

where G = Aut(Js ) is a simple split group scheme of type F4 .

(39.14) Theorem. (1) (Ferrar-Petersson) Let J0 = J(A, »0 ) be a ¬rst Tits con-

struction with A a central simple associative algebra of degree 3. The cohomology

set H 1 F, Aut(J0 , A+ ) classi¬es pairs (J , I ) with J an Albert algebra over F

and I is a central simple Jordan algebra of dimension 9 over F . The cohomology

set H 1 F, Aut(A+ ) classi¬es central simple Jordan algebra of dimension 9 over F .

The sequence of pointed sets

ρ1

ψ

1 ’ F — /NA (A— ) ’ H 1 F, Aut(J0 , A+ ) ’ H 1 F, Aut(A+ )

’ ’

is exact and ψ([»]) = [J(A, »»0 ), A+ ], ρ1 ([J , A ]) = [A ].

(2) Let J be an Albert algebra containing a subalgebra A+ for A central simple of

∼

degree 3. There exist » ∈ F — and an isomorphism φ : J ’ J(A, ») which restricts

’

to the identity on A+ .

(3) J(A, ») is a division algebra if and only if » is not the reduced norm of an

element from A.

Proof : We follow Ferrar-Petersson [?]. (??) We assume for simplicity that F is a

¬eld of characteristic not 2, so that J0 is an F -algebra with a multiplication m.

Let F be the ¬‚ag A+ ‚ J0 and let W = HomF (J0 — J0 , J0 ). We let G = AutF (F)

act on F • W through the natural action. Let w = (0, m). Since AutG (w) =

+

Aut(J0 , A+ ) and since (J0 , A+ )Fsep (J , I )Fsep ( (Js , M3 )Fsep ), the ¬rst claim

follows from (??) and from Corollary (??). The fact that H 1 F, Aut(A+ ) classi¬es

central simple Jordan algebra of dimension 9 over F then is clear.

§39. THE TITS CONSTRUCTION 529

The exact sequence is the cohomology exact sequence associated with the se-

quence (??), where the identi¬cation (??) of F — /NA (A— ) with H 1 F, SL1 (A)

is as follows: let » ∈ F — and let v ∈ A— be such that NAsep (v) = ». Then

sep

± : Gal(Fsep /F ) ’ SL1 (A)(Fsep ) such that ±(g) = v ’1 g(v) is the cocycle induced

by ». The image of the class of » ∈ F — in H 1 F, Aut(J0 , A+ ) is the class of the

cocycle β given by β(g)(a, a1 , a2 ) = a, a1 g(v ’1 )v, v ’1 g(v)a2 . Let γ ∈ GL (J0 )sep

be given by

γ(a, a1 , a2 ) = (a, a1 v ’1 , va2 ),

then β(g) = g(γ ’1 )γ, and, setting J = J(A, »»0 ), one can check that

∼

γ : (J0 , A+ )Fsep ’ (J, A+ )Fsep

’

is an isomorphism, hence (J, A+ ) is the F -form of (J0 , A+ ) given by the image of ».

(??) We set »0 = 1 in (??). Let J be a reduced Freudenthal algebra. By

Theorem (??), we have (J, A+ )Fsep (J0 , A+ )Fsep . Therefore (J, A+ ) is a form of

(J0 , A+ ) and its class in H 1 F, Aut(J0 , A+ ) belongs to the kernel of ρ1 , hence

in the image of ψ. Thus by (??) there exists » ∈ F — such that J(A, »), A+

(J, A+ ), as claimed.

(??), similarly, follows from (??), since J(A, 1) is split (see Example (??),

(??)).

(39.15) Remark. By (??.??), J(A, ») is a division algebra if and only if A is a

division algebra and » is not a reduced norm of A. Examples can be given over

a purely transcendental extension of degree 1: Let F0 be a ¬eld which admits a

division algebra A0 of degree 3 and let A = A0 — F0 (t). Then the Albert algebra

J(A, t) is a division algebra (see Jacobson, [?, p. 417]).

The analogue of Proposition (??) for second Tits constructions is

(39.16) Proposition. Let J0 = J(B, „, u0 , ν0 ) be a second Tits construction and

let „ = Int(u0 ) —¦ „ . The sequence

γ ρ

1 ’ SU(B, „ ) ’ Aut J0 , H(B, „ ) ’ Aut H(B, „ ) ’ 1,

’ ’

where γR (u)(a, b)R = (a, bu’1 )R and ρR is restriction, is exact.

Proof : (??) follows from (??) by descent, using Proposition (??).

To get a result corresponding to Theorem (??) for second Tits constructions, we

recall that the pointed set H 1 F, SU(B, „ ) classi¬es pairs (u, ν) ∈ Sym(B, „ )— —

K — with NB (u) = NK (ν) under the equivalence ≈, where

(39.17)

(u, ν) ≈ (u , ν ) if and only if u = bu„ (b) and ν = ν · NB (b) for some b ∈ B —

(see (??)). As in (??) we set

SSym(B, „ )— = { (u, ν) ∈ H(B, „ ) — K — | NB (u) = NK (ν) }.

(39.18) Theorem. Let J0 = J(B, „, u0 , ν0 ).

(1) The sequence of pointed sets

ψ

1 ’ SSym(B, „ )— /≈ ’

’

ρ1

H 1 F, Aut J0 , H(B, „ ) ’ H 1 F, Aut H(B, „ ) ,

’

530 IX. CUBIC JORDAN ALGEBRAS

where ψ([u, ν]) = [J(B, „, uu0 , νν0 ), H(B, „ )] and ρ1 ([J , A ]) = [A ], is exact.

(2) Let J be a Freudenthal algebra of dimension 27 containing a subalgebra H(B, „ )

for (B, „ ) central simple of degree 3 with a unitary involution. There exist an

∼

admissible pair (u, ν) ∈ SSym(B, „ )— and an isomorphism φ : J ’ J(B, „, u, ν)

’

which restricts to the identity on H(B, „ ).

(3) J(B, „, u, ν) is a division algebra if and only if u is not the reduced norm of an

element from B — .

Proof : The proof of (??) is similar to the one of Theorem (??) and we skip it.

Any Hurwitz algebra can be obtained by successive applications of the Cayley-

Dickson process, starting with F . The next result, which is a special case of a

theorem of Petersson-Racine [?, Theorem 3.1], shows that a similar result holds

for Freudenthal algebras of dimension 3, 9 and 27 if Cayley-Dickson processes are

replaced by Tits processes:

(39.19) Theorem (Petersson-Racine). Assume that char F = 3. Any Freudenthal

algebra of dimension 3, 9 or 27 can be obtained by successive applications of the

Tits process. In particular any exceptional Jordan algebra of dimension 27 is of the

form H(B, „ ) • B where B is a central simple of degree 3 over a quadratic ´tale

e

F -algebra K with a unitary involution.

Proof : A Freudenthal algebra of dimension 3 is a cubic ´tale algebra, hence the

e

claim follows from Example (??), (??), if dim J = 3. The case dim J = 9 is

covered by Example (??), (??). If J has dimension 27 and J contains a Freudenthal

subalgebra of dimension 9 of the type H(B, „ ), then by Theorem (??), there exists

a pair (u, ν) such that J (B, „, u, ν). Thus we are reduced to showing that J

contains some H(B, „ ). If J is reduced this is clear, hence we may assume that J

is not reduced. Then (see the proof of (??)) F is an in¬nite ¬eld. Let L be a cubic

´tale F -subalgebra of J and let J = L • V , V = (V, L, Q, β), be the corresponding

e

Springer decomposition. For some v ∈ V , the set {v, β(v)} is linearly independent

over L since Q v, β(v) is anisotropic and by a density argument (F is in¬nite) we

may also assume that Q restricted to U = Lv • Lβ(v) is L-nonsingular. Thus J

contains a Springer construction J1 = J(L, U ) of dimension 9. In view of the 9-

dimensional case J1 is a Tits construction and by Example (??), (??), J1 H(B, „ )

for some central simple algebra (B, „ ) of degree 3 with unitary involution, hence

the claim.

Jordan algebras of the form L+ (L cubic ´tale of dimension 3), A+ (A central

e

simple of degree 3), or H(B, „ ) (B central simple of degree 3 with an involution

of the second type) are “generic subalgebras ” of Albert algebras in the following

sense:

(39.20) Proposition. Let J be an Albert algebra.

(1) There is a Zariski-open subset U of J such that the subalgebra generated by x

is ´tale for all x ∈ U .

e

(2) There is a Zariski-open subset U of J such that the subalgebra generated by

x ∈ U and y ∈ U is of the form A+ , for A central simple of degree 3 over F , or of

the form H(B, „ ) for B central simple over a quadratic separable ¬eld extension K

and „ a unitary involution.

Proof : The ¬rst claim is already in (??), (??). The second follows from the proof

of (??).

§40. COHOMOLOGICAL INVARIANTS 531

(39.21) Remark. The element v in the proof of (??) is such that v and β(v) are

linearly independent over L (see Proposition (??)). If v is such that β(v) = »v for

» in L, then J is reduced by Theorem (??) and (L, v) generates a 6-dimensional

subalgebra of J of the form H3 (F, ±) (Soda [?, Theorem 2]). Such an algebra is

not “generic”.

(39.22) Remark. If char F = 3, the only Freudenthal algebras which cannot be

obtained by iterated Tits constructions are separable ¬eld extensions of degree 3

(see [?, Theorem 3.1]). We note that Petersson and Racine consider the more

general case of simple cubic Jordan structures (not just Freudenthal algebras) in

[?, Theorem 3.1].

The Albert algebra Js = J M3 (F ), 1 is split, thus G = Aut J M3 (F ), 1 is

a simple split group scheme of type F4 .

(39.23) Proposition. The pointed set H 1 F, (GL3 — GL3 )Det / Gm Z/2Z clas-

si¬es pairs (J, H B, „ ) where J is an Albert algebra, B ‚ J is central simple with

unitary involution „ over a quadratic ´tale algebra K. The map

e

H 1 F, (GL3 — GL3 )Det / Gm Z/2Z ’ H 1 (F, G),

induced by AutF J M3 (F ), 1 , M3 (F )+ ’ AutF J M3 (F ), 1 and which asso-

ciates the class of J to the class of J, H(B, „ ) is surjective.

Proof : The ¬rst claim follows from Corollary (??) and Theorem (??), the second

then is a consequence of Theorem (??).

(39.24) Remark. Let J be an Albert algebra. We know that J J(B, „, u, β)

for some datum (B, „, u, ν). The datum can be reconstructed cohomologically as

follows. Let [±] ∈ H 1 F, (GL3 — GL3 )Det / Gm Z/2Z be a class mapping to [J].

The image [γ] ∈ H 1 (F, Z/2Z) of [±] under the map in cohomology induced by

the projection (GL3 — GL3 )Det / Gm Z/2Z ’ Z/2Z de¬nes the quadratic exten-

sion K. Pairs (J, H B, „ ) with ¬xed K are classi¬ed by

H 1 F, (GL3 — GL3 )Det / Gm γ

and the projection on the ¬rst factor gives an element of H 1 F, (PGL3 )γ , hence

by Remark (??) a central simple K-algebra B with unitary involution „ . We ¬nally

get (u, ν) from the exact sequence (??.??).

§40. Cohomological Invariants

In this section we assume that F is a ¬eld of characteristic not 2. Let J =

H3 (C, ±) be a reduced Freudenthal algebra of dimension > 3. Its bilinear trace

form is given by

T = 1, 1, 1 ⊥ bC — ’b, ’c, bc

where bC is the polar of NC . As known from Corollary (??) and Theorem (??),

the P¬ster forms bC and bC — b, c determine the isomorphism class of J. Let

dimF C = 2i , let fi (J) ∈ H i (F, Z/2Z) be the cohomological invariant of the

P¬ster form bC and let fi+2 (J) ∈ H i+2 (F, Z/2Z) be the cohomological invariant

of bC — b, c . These two invariants determine J up to isomorphism. Observe

that Freudenthal algebras of dimension 3 with zero divisors are also classi¬ed by a

cohomological invariant: Such an algebra is of the form F + — K + and is classi¬ed

532 IX. CUBIC JORDAN ALGEBRAS

by the class of K f1 (K) ∈ H 1 (F, S2 ). We now de¬ne the invariants f3 (J) and

f5 (J) for division algebras J of dimension 27 (and refer to Proposition (??), resp.

Theorem (??) for the corresponding invariants of algebras of dimension 3, resp. 9).

We ¬rst compute the bilinear trace form of J.

(40.1) Lemma. (1) Let (B, „ ) be a central simple algebra of degree 3 over a quad-

ratic ´tale F -algebra K with a unitary involution and let T„ be the bilinear trace

e

form of the Jordan algebra H(B, „ ). Then

for b, c ∈ F —

T„ 1, 1, 1 ⊥ bK/F — ’b, ’c, bc

where bK/F stands for the polar of the norm of K.

(2) Let J be a Freudenthal algebra of dimension 27 and let T be the bilinear trace

form of J. There exist a, b, c, e, f ∈ F — such that

T 1, 1, 1 ⊥ 2 — a, e, f — ’b, ’c, bc .

Proof : (??) follows from Proposition (??).

(??) By Theorem (??) we may assume that J is a second Tits construction

J(B, „, u, µ), so that

T (x, y), (x , y ) = T„ (x, x ) + TK/F TrdB yu„ (y )

for x, x ∈ H(B, „ ) and y, y ∈ B. By Lemma (??), (??), we may assume that

NrdB (u) = 1. Let „ = Int(u’1 ) —¦ „ . By (??) the trace form of H(B, „ ) is of the

form

T„ = 1, 1, 1 ⊥ bK/F — ’e, ’f, ef

for e, f ∈ F — . Let T„,„ (x, y) = TK/F TrdB xu„ (y) for x, y ∈ B. We claim that

T„,„ bK/F — ’b, ’c, bc — ’e, ’f, ef .

The involution „ is an isometry of the bilinear form T„,„ with the bilinear form

(TK/F )— (TB,„,u ) where TB,„,u (x, y) = TrdB „ (x)uy . Thus it su¬ces to have an

isomorphism of hermitian forms

TB,„,u ’b, ’c, bc —K ’e, ’f, ef

K K

since

(TK/F )— ±1 , . . . ±n = bK/F — ±1 , . . . ±n .

K

In view of Proposition (??) the unitary involution „ — „ on B —K ι B corre-

sponds to the adjoint involution on EndK (B) of the hermitian form T(B,„,u) under

the isomorphism „— : B —K ι B ’ EndK (B). By the Bayer-Lenstra extension (??)

of Springer™s theorem, we may now assume that B = M3 (K) is split, so that by

Example (??) „ is the adjoint involution of ’b, ’c, bc K and „ is the adjoint invo-

lution of ’e, ’f, ef K . This shows that TB,„,u and ’b, ’c, bc K —K ’e, ’f, ef K

are similar hermitian forms and it su¬ces to show that they have the same deter-

minant. By Corollary (??) the form T(B,„,u) has determinant the class of Nrd(u),

which, by the choice of u, is 1. Thus we get

T T„ ⊥ T„,„ 1, 1, 1 ⊥ 2 — a, e, f — ’b, ’c, bc

√

where K = F ( a).

§40. COHOMOLOGICAL INVARIANTS 533

(40.2) Theorem. (1) Let F be a ¬eld of characteristic not 2. For any Freudenthal

algebra J of dimension 3 + 3 · 2i , 1 ¤ i ¤ 3, there exist cohomological invariants

fi (J) ∈ H i (F, Z/2Z) and fi+2 (J) ∈ H i+2 (F, Z/2Z) which coincide with the invari-

ants de¬ned above if J is reduced.

(2) If J = J(B, „, u, ν) is a second Tits construction of dimension 27, then f3 (J)

is the f3 -invariant of the involution „ = Int(u) —¦ „ of B.

√

Proof : (??) Let K = F ( a). With the notations of Lemma (??), the invariants are

given by the cohomological invariants of the P¬ster forms a , resp. a, b, c if J

has dimension 9 and the cohomological invariants of a, e, f , resp. a, e, f — b, c

if J has dimension 27. The fact that these are fi -, resp. fi+2 -invariants of J follows

as in Corollary (??).

Claim (??) follows from the computation of T„ .

(40.3) Corollary. If two second Tits constructions J(B, „, u1 , ν1 ), J(B, „, u2 , ν2 )

of dimension 27 corresponding to di¬erent admissible pairs (u1 , ν1 ), (u2 , ν2 ) are

isomorphic, then there exist w ∈ B — and » ∈ F — such that »u2 = wu1 „ (w). If

furthermore Nrd(u1 ) = Nrd(u2 ), then w can be chosen such that u2 = wu1 „ (w).

Proof : Let „i = Int(ui ) —¦ „ , i = 1, 2. In view of Theorem (??), (??), and

Theorem (??), the involutions „1 and „2 of B are isomorphic, hence the ¬rst

claim. Taking reduced norms on both sides of »u2 = wu1 „ (w) we get »3 =

’1

NrdB (w) NrdB „ (w) and » is of the form » » . Replacing w by w» , we get

the second claim.

(40.4) Remark. Corollary (??) is due to Parimala, Sridharan and Thakur [?].

As we shall see in Theorem (??) (which is due to the same authors) w can in fact

be chosen such that u2 = wu1 „ (w) and ν2 = ν1 NrdB (w) so that the converse of

Lemma (??) holds.

For an Albert algebra J with invariants f3 (J) and f5 (J), the condition f3 (J) =

0 obviously implies f5 (J) = 0. More interesting are the following two propositions:

(40.5) Proposition. Let J be an Albert algebra. The following conditions are

equivalent:

(1) J is a ¬rst Tits construction, J = J(A, »).

(2) There exists a cubic extension L/F such that JL splits over L.

(3) The Witt index w(T ) of the bilinear trace form T of J is at least 12.

(4) f3 (J) = 0.

(5) For any Springer decomposition J = J(V, L) with corresponding twisted com-

position “ = (V, L, Q, β), we have wL (Q) ≥ 3.

Proof : (??) ’ (??) Choose L which splits A.

(??) ’ (??) By Springer™s Theorem we may assume that J is split. The claim

then follows from the explicit computation of the bilinear trace form given in (??).

(??) ’ (??) We have

T 1, 1, 1 ⊥ a, e, f — ’b, ’c, bc .

Thus, if w(T ) ≥ 12, the anisotropic part ban of a, e, f — ’b, ’c, bc has at most

dimension 6; since ban ∈ I 3 F , the theorem of Arason-P¬ster (see Lam [?, p. 289])

shows that ban = 0 in W F . Lemma (??) then implies that a, e, f is hyperbolic,

hence f3 (J) = 0.

534 IX. CUBIC JORDAN ALGEBRAS

(??) ’ (??) Let J = J(V, L) be a Springer decomposition of J for a twisted